can you list the steps on how to solve this (how to get to the answer), please. ---### Calculus Problem: Related Rates**Problem Statement:**The length of a rectangle is increasing at a rate of $6 \, \text{cm/s}$ and its width is increasing at a rate of $3 \, \text{cm/s}$. When the length is $13 \, \text{cm}$ and the width is $9 \, \text{cm}$, how fast is the area of the rectangle increasing?\[ \text{Rate of increase of area} = \boxed{ \text{cm}^2/\text{s} } \]---**Explanation:**This problem involves related rates, a common type of problem in calculus where we determine how one quantity changes as another quantity changes. To solve this problem, you must understand how the dimensions of a geometric shape (in this case, a rectangle) relate to its area and how changes in these dimensions affect the area over time.1. **Identify Variables:** - Let $ l $ represent the length of the rectangle. - Let $ w $ represent the width of the rectangle. - Let $ A $ represent the area of the rectangle.2. **Given Rates:** - The rate of change of length $ \frac{dl}{dt} = 6 \, \text{cm/s} $ - The rate of change of width $ \frac{dw}{dt} = 3 \, \text{cm/s} $3. **Formula for the Area of Rectangle:** - The area of the rectangle is $ A = l \times w $4. **Differentiate with Respect to Time:** - Use the product rule to differentiate the area with respect to time. \[ \frac{dA}{dt} = \frac{d}{dt} (l \times w) = l \frac{dw}{dt} + w \frac{dl}{dt} \]5. **Substitute Given Values:** - Length $ l = 13 \, \text{cm} $ - Width $ w = 9 \, \text{cm} $ - Now plug in the values and the rates of change into the differentiated formula. \[ \frac{dA}{dt} = (13 \, \text{cm}) \times (3 \, \text{cm/s})

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Answered: n you list the steps on how to s

Math

Advanced Math

n you list the steps on how to s

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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